\section{Digital Option}

Consider a digital option. This option pays 1 if the stock price is above
certain values (strike) and zero otherwise.\\
In a risk neutral world the probability of an asset being above the strike
price at the maturity of an option is $\Phi(d_2)$ with $\Phi$ the probability
density function of a normal distribution and $d_2=d_1-\sigma\sqrt T$ and $d_1
=\frac{log(S_0/K) + (r+\sigma^2/2)T}{\sigma\sqrt T}$. Therefore the analytical
value of a digital option is equal to $e^{-rT}\Phi(d_2)$.\\
This option price can be approximated with the help of Finite Difference, but
normally the payoff of a digital option is not differentiable. In order to
solve this problem we approximate the digital option with the normal
cumulative distribution function. The option price is approximated in an
optimal way if the standard deviation of the normal cdf is taken as small as possible.
However, as will be shown, the minimum value of the standard deviation is dependent on
the coarseness of the grid. The normal cdf is of the following form
\[
\frac{1}{2} \left(1+\mathrm{erf}\left( \frac{x-\mu}{\sigma\sqrt2}\right) \right),
\]
in which $\mu$ should be equal to the strike price of the option and $\sigma$
should be taken very small to get a differentiable variant of the digital
option.\\
\newline
\noindent
Figure \ref{fig:digitalsd} shows the influence of different values of the
standard deviation on the approximation of the premium of the digital option.
From the left figure can be concluded that the size of $\Delta x$ is not
influencing the approximation of the premium of the digital option. This in
contrast to the results from the left figure. In this figure it is clear that
using a cdf with a standard deviation of 0.001 causes the pay off to ill
behave at the strike price (see the green spikes around S = 110 and Premium
0.5). This shows that when approximating the premium of a digital option using
finite difference, the standard deviation should not be taken too small in
respect to the coarseness of the grid, since this may lead to ill behaving
solutions.


\begin{figure}[htbp]
\begin{center}
\caption{Digital option with different standard deviations}\label{fig:digitalsd}
\includegraphics[width=7cm]{digital_dx_001}
\includegraphics[width=7cm]{digital_dx_00001}
\end{center}
{\footnotesize Values of $X_{min}=1$ and $X_{max}=3$ were used to get these
figures.}
\end{figure}

The choice of the standard deviation not only influences the quality of the
approximation of the premium, it also affects the correctness of the
approximation of the $\Delta$. In the right sub figure of figure
\ref{fig:digitaldx}  shows that when using a very fine grid with a small
standard deviation the approximation of the $\Delta$ can be incorrect (see the
green spikes in the left figure and the detail in the right figure). This
stresses the importance of choice of the variables in finite difference.

\begin{figure}[htbp]
\begin{center}
\caption{Digital option for different values of $\Delta x$}\label{fig:digitaldx}
\includegraphics[width=7cm]{delta_digital_full}
\includegraphics[width=7cm]{delta_digital_detail}
\end{center}
{\footnotesize The $\Delta x$ in this plot is 0.0001}
\end{figure}
